PipeModel Idealized valley microclimate sandbox

with robust modeling, spatial CV, and land-cover physics

Author

gisma

1 Why the PipeModel?

The PipeModel is a deliberately idealized yet physically plausible valley scenario. It distills terrain to the essentials (parabolic cross-valley profile) and optional features (left-side hill, right-side pond or hollow), so that dominant microclimate drivers become visible and testable:

  • Radiation via terrain exposure cos(i) from slope & aspect
  • Elevation: daytime negative lapse; pre-dawn weak inversion
  • Cold-air pooling along the valley axis (Gaussian trough)
  • Surface type / land-cover (grass / forest / water / bare soil / maize) alters heating, shading, roughness and nocturnal behaviour

You can sample synthetic stations, train interpolators (IDW, Kriging variants, RF, GAM), and assess them with spatial LBO-CV.

🔧 This document keeps the previous behaviour but extends the physics with a modular land-cover layer that feeds into both daytime and night fields.

2 D. Physics & Scenario Builder — Cheat Sheet (enhanced LC model)

2.1 D.1 Generated rasters & derived fields

Name Unit What it is How it’s built
E (elev) m Ground elevation Parabolic “half-pipe” across y; + optional hill; − optional pond/hollow
slp, asp rad Slope, aspect terra::terrain(E, "slope"/"aspect", "radians")
I14, I05 – Cosine solar incidence at 14/05 UTC cosi_fun(alt, az, slp, asp), clamped to [0,1]
lc cat Land-cover class {Forest, Water, Bare Soil, Maize}; rules from hill/slope/water masks
hillW 0–1 Hill weight (1 inside footprint) Disk/Gaussian on left third; combines main + optional micro-hills
lake 0/1 Water mask 1 only when lake_choice == "water" (disk on right third)
I14_eff – Shaded incidence (day) I14 * shade_fac_by_lc[lc]
αI(lc) – Daytime solar sensitivity by LC Look-up from alpha_I_by_lc
dawn_bias(lc) °C Additive pre-dawn bias by LC Look-up from dawn_bias_by_lc
pool_fac(lc) – Pooling multiplier by LC Look-up from pool_fac_by_lc
R14 (T14) °C Daytime “truth” temperature field Eq. (below)
R05 (T05) °C Pre-dawn “truth” temperature field Eq. (below)

2.2 D.2 Governing equations

Let $$ be the domain-mean elevation. Define the cross-valley cold-pool kernel

\[ \texttt{pool\_base} \;=\; A \exp\!\left[-(d_y/w)^2\right],\quad d_y=|y-y_0|, \]

blocked over the hill by (1 − pool_block_gain * hillW).

Day (14 UTC)

\[ T_{14} \;=\; T0_{14} \;+\; \texttt{lapse\_14}\,(E-\overline{E}) \;+\; \alpha_I(\texttt{lc})\, I_{14}^{\text{eff}} \;+\; \varepsilon_{14}, \quad I_{14}^{\text{eff}} = I_{14}\cdot \texttt{shade\_fac}(\texttt{lc}). \]

Pre-dawn (05 UTC)

\[ T_{05} \;=\; T0_{05} \;+\; \texttt{inv\_05}\,(E-\overline{E}) \;+\; \eta_{\text{slope}}\;\texttt{slp} \;-\; \texttt{pool\_base}\cdot(1-\texttt{pool\_block\_gain}\cdot\texttt{hillW})\cdot \texttt{pool\_fac}(\texttt{lc}) \;+\; \texttt{dawn\_bias}(\texttt{lc}) \;+\; \varepsilon_{05}. \]

Noise $_{14},_{05} (0,,0.3^2)$ i.i.d.

Note vs. predecessor: the former warm_bias_water_dawn * lake term is now folded into dawn_bias(lc) (class “Water”); daytime α_map became αI(lc) * I14_eff with explicit canopy shading.

2.3 D.3 Dials

2.3.1 Global scalars

Parameter Default Sensible range Affects Visual signature (+)
T0_14 26.0 °C 20–35 T14 baseline Uniform warming
lapse_14 −0.0065 °C/m −0.01…−0.002 T14 vs elevation Cooler rims, warmer floor
T0_05 8.5 °C 3–15 T05 baseline Uniform warming
inv_05 +0.003 °C/m 0–0.008 T05 vs elevation Rims warmer vs floor
η_slope 0.6 0–1.5 T05 slope flow proxy Steeper slopes a bit warmer at dawn
pool_base amplitude 4.0 K 1–8 T05 pooling depth Stronger blue band on valley axis
w_pool 70 m 40–150 T05 pooling width Narrower/broader cold band
pool_block_gain 0.4 0–1 Hill blocking Warm “tongue” over hill at dawn
noise σ 0.3 K 0–1 Both Fine speckle/random texture

2.3.2 Land-cover coefficients (by class)

Defaults used in the code:

LC class alpha_I_by_lc shade_fac_by_lc dawn_bias_by_lc (°C) pool_fac_by_lc
Forest 3.5 0.6 +0.3 0.7
Water 1.5 1.0 +1.2 0.8
Bare Soil 6.0 1.0 −0.5 1.1
Maize 4.5 0.9 +0.1 1.0

Interpretation: Bare Soil heats most by day and enhances pooling (factor > 1) and cool bias at dawn; Forest damps radiation by day (shading) and reduces pooling (factor < 1); Water heats little by day, gets a positive dawn bias and reduced pooling; Maize sits between grass and forest.

2.3.3 Geometry/toggles

Parameter Default Options / range Effect
lake_choice "water" "none", "water", "hollow" Controls depression; only "water" sets LC=Water (thermal effects).
hill_choice "bump" "none", "bump" Adds blocking & relief.
lake_diam_m 80 40–150 Size of pond/hollow.
lake_depth_m 10 5–30 Depression depth.
hill_diam_m 80 40–150 Hill footprint.
hill_height_m 50 10–120 Hill relief.
smooth_edges FALSE bool Soft pond rim if TRUE.
hill_smooth FALSE bool Gaussian hill if TRUE.
(optional) micro-hills off random_hills, micro_* Adds sub-footprint relief; included in hillW.

2.4 D.4 Quick “recipes”

  • Cloud/haze day → ↓ alpha_I_by_lc (all classes, esp. Bare/Maize) → daytime LC contrasts fade; models lean on elevation/smoothness.
  • Hotter afternoon → ↑ T0_14 (+1…+3 K) → uniform bias shift; rankings unchanged.
  • Stronger pooling → ↑ pool_base and/or ↓ w_pool → sharper, deeper trough; drift-aware models gain.
  • Water vs hollow → "water" sets LC=Water → ↓ daytime heating, ↑ dawn warm bias, ↓ pooling; "hollow" keeps only geometry (no water thermals).
  • Hill blocking → ↑ pool_block_gain → warm dawn tongue over hill; harder CV across blocks.
  • Cover swaps (what if): set a patch to Bare Soil → warmer day, colder dawn & stronger pooling; to Forest → cooler day, weaker pooling & slight dawn warm-up.

2.5 Scaled demo: Compact Physics Dossier

Here’s a clear, didactic walkthrough of the “scaled teaching” scenario and exactly what R_true14 and R_true05 are.

2.6 Lake–Bump–Dense: Compact Physics Dossier

Goal. A clear, didactic synthetic scenario that (a) looks realistic, (b) drives temperature with topography + land-cover + sun, and (c) plays nicely with blocked CV and R*-tuning. Class 4 is meadows (not maize).

2.7 G Diagnostic

let’s read your baseline (no R*) results explicitly through the lens of process (what drives T) and scale (over what distances the drivers operate), model-by-model and time-by-time, then close with a scale+process summary and concrete upgrades.

3 T14 (daytime)

Process you’re trying to capture

  • Shortwave forcing projected by slope/aspect → very local facet contrasts.
  • Land-cover (LC) modulates heating (forest shade, water inertia) at patch scale.
  • A mild negative lapse with elevation (broad scale).
  • Anisotropy is limited; key is small-scale facet/LC contrasts.

Observed performance (LBO-CV) RMSE ↓ / R² ↑: GAM (0.436 / 0.642) < KED (0.446 / 0.630) ≈ RF (0.449 / 0.619) ≪ IDW (0.813 / 0.060), Voronoi (0.828 / 0.025), OK (0.848 / 0.085). Bias is small for the top 3 (GAM +0.032, KED +0.014, RF +0.050 °C).

3.0.1 What the diagnostics mean model-by-model

  • GAM — best alignment to process and scale

    • Boxplots: tight across blocks → it’s matching facet/patch scales.
    • Obs–Pred: near 1:1 with mild underfit only at the hottest facets.
    • Residual density: narrow, centered at ~0 → low variance, low bias.
    • Why: smooth terms over cos(i), slope, z, LC let it bend at the right (small) scales without oversmoothing.

  • KED — close second but still smoothing across LC edges

    • Boxplots: slightly wider tails in blocks crossing LC transitions.
    • Obs–Pred: more scatter than GAM; extremes compressed a bit.
    • Residual density: centered but broader.
    • Why (scale): isotropic variogram + untuned drift scale → blurs patch edges. You need LC as drifts and R*-smoothed topography terms.

  • RF — competitive third; sensitive to micro-texture

    • Boxplots: a tad broader tails → some patchy flicker in blocks.
    • Obs–Pred: good alignment; small warm bias (+0.05 °C).
    • Residual density: narrow, near-zero mean.
    • Why (scale): with raw x,y and unsmoothed features it can pick up too-fine structure; it still handles LC×cos(i) nonlinearity well.

  • OK / IDW / Voronoi — scale/process mismatch

    • Boxplots: wide with outliers → leakage across sharp contrasts.
    • Obs–Pred: under-dispersion (slope < 1 feel), big scatter.
    • Residual density: broad / skewed.
    • Why: purely spatial kernels ignore physics; their smoothing scale is wrong for facet/patch structure.

Day takeaway: day is short-scale, LC-modulated. Models that encode that structure (GAM, RF) win; kriging needs right drifts at the right scale to catch up.

4 T05 (pre-dawn)

Process you’re trying to capture

  • Cold-air pooling: a cross-valley trough (short scale across, longer along → anisotropy).
  • Slope term (drainage tendency).
  • LC offsets (water warmest, bare coolest) and small inversion with elevation.

Observed performance (LBO-CV) RMSE ↓ / R² ↑: RF (0.434 / 0.939) < GAM (0.622 / 0.864) ≪ KED (0.900 / 0.707) < OK (1.121 / 0.547) < Voronoi (1.271 / 0.440) < IDW (1.457 / 0.246). Bias: RF −0.018 (tiny cool), GAM −0.005, KED +0.106 (under-cools), IDW −0.137 (over-cools).

4.0.1 What the diagnostics mean model-by-model

  • RF — clear winner; nails nonlinear pooling+LC

    • Boxplots: tightest by far → right scale and good generalization.
    • Obs–Pred: almost exactly 1:1 → calibrated.
    • Residual density: slim, centered slightly negative (~−0.02 °C).
    • Why (process): tree splits capture trough + slope + LC interactions; less sensitive to isotropy assumptions.

  • GAM — strong second; smooth but misses sharp minima

    • Boxplots: tight but a bit wider than RF on trough blocks.
    • Obs–Pred: close to 1:1; modest extra spread.
    • Residual density: centered, slightly wider than RF.
    • Why (scale): splines smooth; without R*-tuned features they can round off the deepest pooled cold.

  • KED — middle of the pack; wrong mean for pooling

    • Boxplots: broader with tails in trough/blocked-flow blocks.
    • Obs–Pred: under-dispersion; misses deep minima.
    • Residual density: shifted positive (+0.106 °C) → under-cooling.
    • Why (process & anisotropy): elevation drift ≠ pooling; variogram likely isotropic, so it leaks across the cross-valley gradient. Needs distance-to-axis, cross-valley coordinate, hill-block mask, and anisotropic variogram.

  • OK / Voronoi / IDW — struggle in anisotropic pooling

    • Boxplots: very wide; many outliers → big scale mismatch.
    • Obs–Pred: noisy; IDW shows global over-cool bias.
    • Residual density: broad (IDW skewed negative).
    • Why: they smooth across the short cross-valley scale and ignore LC offsets.

Night takeaway: night is anisotropic and thresholdy. RF handles that best; GAM is close with proper feature scale. Kriging must get the mean field right and adopt directional scale to compete.

5 Scale & process, integrated (what each model is buying/missing)

Time Model What process it encodes How it treats scale What the metrics+plots say
T14 GAM cos(i) × LC × z interactions (smooth) Implicit via spline basis; good at patch/facet Best RMSE/R²; tight boxes; slender residuals → matched to small scales
T14 RF Nonlinear LC × cos(i) well; can chase micro-texture Learns whatever scale is in features (and x,y) Near-best metrics; slightly broader boxes → feature scale not tuned
T14 KED Mean = linear drifts (z, slope, cosi, maybe LC) Variogram smooths across LC edges Good but behind GAM; tails at LC transitions
T14 OK/IDW/Voro None Kernel/variogram at one scale Broad tails, under-dispersion → process blind
T05 RF Pooling trough + slope + LC (thresholdy) Chooses effective scales from features Top RMSE/R², clean calibration; best boxes/density
T05 GAM Smooth trough + slope + LC offsets Smooths; needs tuned features Strong second; misses sharp minima a bit
T05 KED Wrong mean for pooling if only z/slope Variogram often isotropic Warm bias (+0.106 °C), broad boxes → needs pooling drifts & anisotropy
T05 OK/IDW/Voro None One isotropic smoothing scale Very wide tails; density broad/skewed

6 What to change (small steps, big benefits)

1) Add the missing process to kriging (both times)

  • Day: include cos(i) and LC dummies as external drifts; compute cos(i) from the actual sun.
  • Night: add cross-valley coordinate / distance-to-axis, a hill-block mask, and LC offsets as drifts.
  • This makes KED’s mean physically right; the variogram only cleans residual texture.

2) Match the scale of the features (R*)

  • For z, slope, cos(i), scan R over a practical range (e.g., variogram L50→L95) with blocked CV and rebuild features at R*.
  • Expect narrower boxplots and slimmer residual densities for GAM (T14) and RF (T05); KED gains a lot too.

3) Respect anisotropy at night

  • Rotate to (s,t) (along/cross-valley); give shorter range in t for variograms.
  • Even without an explicit anisotropic variogram, feeding t as a drift and smoothing features at R* helps.

4) Hybridize: regression-kriging

  • Mean = GAM (T14) / RF (T05); residuals = OK/KED with short-range, anisotropic structure.
  • Keeps the physics-savvy mean and mops up local spatial leftovers.

5) RF hygiene (avoid coordinate memorization)

  • Drop raw x,y or replace with oriented (s,t); rely on R*-smoothed z/slope/cos(i) + LC and pooling drifts.
  • This keeps process, reduces overfitting to station layout.

6) Validation remains scale-aware

  • Keep LBO; try a few random grid origins (tiling jitter) and confirm ranks stay stable.

6.1 Summary

Daytime temperature is controlled by very local facet and LC effects layered over a gentle lapse; models that encode those drivers at the right (small) scale—notably GAM, then RF—generalize across blocks with low error.

Pre-dawn temperature is anisotropic with a short cross-valley pooling scale, slope, and LC offsets; RF captures these thresholdy interactions best, with GAM second. Purely spatial smoothers (OK/IDW/Voronoi) underperform because their smoothing scale and mean process are mismatched.

Bring kriging back into contention by giving it the right drifts (cos(i), LC, distance-to-axis, hill-block) at tuned feature scales (R*), and by acknowledging anisotropy at night; if you want the best of both worlds, use regression-kriging with the learned mean from GAM/RF and an anisotropic residual field.

7 Critical review: does the winner take it all?

Short answer: no. Even though the baseline shows GAM (day) and RF (pre-dawn) leading on block-CV, a “winner-takes-all” policy is brittle because:

  • Regime shifts: Day vs. night, clear vs. cloudy, dry vs. wet canopy, snow, leaf-on/off—each changes the dominant process and therefore the right scale. Your “winner” can flip.
  • Sampling artifacts: With a different station layout or fewer stations, RF can overfit locations; kriging can swing with a refit variogram; GAM can underfit sharp minima if features aren’t scale-tuned.
  • Extrapolation: RF/GAM extrapolate poorly beyond the feature envelope (new hill, bigger lake). Kriging extrapolates linearly in the drift but may oversmooth. The best model by CV is not always the safest out-of-sample.
  • Uncertainty: Kriging gives a variance; RF/GAM need extra work (quantile/ensembles) for predictive intervals. If you “winner take all,” you may lose calibrated uncertainty where you need it most.

8 “Information bias” between models

Different learners consume different information channels and bring their own priors. That creates systematic biases you can anticipate and manage.

Model Preferred info Built-in bias Typical failure mode
Voronoi/IDW Distance only to stations Locality bias; no physics Edge artefacts; oversmooth across LC boundaries; anisotropy ignored
OK Distance + stationarity (residual field) Global smoothing scale; isotropy unless told otherwise Under-dispersion of extremes; leakage across cross-valley trough
KED OK + drifts (z, slope, cos(i), LC) Mean = whatever drifts encode; scale of drift matters If drift misses physics (pooling), mean is wrong → biased; if drift scale is off → blur
GAM Smooth functions of features (z, slope, cos(i), LC) Smoothness bias; picks a scale implied by basis Rounds off sharp minima/maxima if features aren’t R*-tuned
RF Nonlinear interactions in features; can use x,y Sample-density & coordinate bias (memorization) Patchy “salt-and-pepper”; poor extrapolation; learns layout if x,y left in

How to reduce these biases

  • RF: remove raw x,y (or replace with oriented s,t), feed R*-smoothed z/slope/cos(i) + explicit pooling/LC drifts → makes it learn process, not positions.
  • GAM: ensure R* on features so the spline’s smoothness matches the process scale.
  • KED/OK: add the right drifts (cos(i)@day; distance-to-axis, hill-block, LC@night) and consider anisotropic variograms or rotated coords.

9 What the current results imply (winner vs. information bias)

Day (T14)

  • GAM wins because it converts facet + LC physics into smooth effects at the correct small scales. Bias watch: will under-hit extremes if features are raw/noisy → fix with R*.
  • RF close; if x,y are present or features are too fine, it may overfit micro-texture. Mitigation: drop x,y; use R* features.
  • KED behind because the drift/variogram combo blurs LC edges; give it cos(i)+LC drifts and R* to recover.

Pre-dawn (T05)

  • RF wins by capturing pooling×slope×LC interactions (thresholdy, anisotropic). Bias watch: if station layout changes, performance can drift—guard with spatial CV and no x,y.
  • GAM close but smooths the deepest minima unless features reflect the trough’s short cross-valley scale → tune R*.
  • KED/OK underperform without an explicit pooling drift and anisotropy; that’s information bias: they’re limited by what you tell the mean and by isotropic smoothing.

10 Don’t pick one—blend them (practical recipe)

  1. Regime-aware mean
  • Use GAM for T14, RF for T05 means (after R* tuning and with physics features).
  • Remove x,y from RF; use (s,t) if you need location signals.
  1. Residual kriging
  • Krige residuals from the mean with a short-range, anisotropic variogram (short across-valley, longer along-valley). This adds local spatial coherence and gives an uncertainty surface.
  1. Stacking with block-CV
  • Train a simple meta-learner on out-of-block predictions (GAM, RF, KED) → get weights that vary by time/regime.
  • Or per-block weights: \(w_m(b) \propto 1/\text{RMSE}_{m,b}\), then blend predictions inside each block and smooth the weights.
  1. Agreement/diagnostic maps
  • Export disagreement maps (max–min across models) and which-model-won maps per block/time. High disagreement = low trust areas.
  1. Uncertainty
  • Keep kriging variance from residual-OK. For RF, add quantile forest; for GAM, use posterior SE as a rough guide (not predictive). Report a combined interval (mean ± kriging SD ⊕ model spread).

11 Bottom line

  • The current leaders (GAM@day, RF@night) deserve their spots—they align best with the dominant processes and scales.
  • But each model carries information bias (smoothness, stationarity, coordinate focus) that will bite under layout changes, regime shifts, or extrapolation.
  • Replace “winner takes all” with a process-aware ensemble: R*-tuned features, regime-specific mean (GAM/RF), anisotropic residual kriging, and CV-weighted stacking.
  • Always publish a skill map, a disagreement map, and uncertainty—that’s how you turn a good score into a reliable microclimate product.

11.1 I. Scale analysis — L50/L95 & tuned KED drift (R*)

This section adds a four-stage pipeline:

  1. Scale inference: global variogram → L50/L95
  2. Scale-matched predictors: drift from smoothed E at radius R
  3. Tune R* with blocked CV (U-curve)
  4. Diagnostics: full benchmark + simple error budget

Why: Matching the model scale to the process scale reduces scale-mismatch error and makes gains attributable to scale rather than algorithm choice.

11.1.1 Reading the outputs

  • Variogram: dotted sill; dashed L50/L95 → scale anchors for smoothing and block sizes.
  • U-curve: R* at lowest blocked-CV RMSE; include R = 0 so the tuner can prefer the raw drift.
  • Benchmark: compare OK / KED / GAM / RF / IDW / Voronoi under the same blocked CV; document block size and R*.
  • Error budget (illustrative): OK → KED(base) → KED(R*) shows gains from drift and from scale matching.

From concept to practice (pipeline mapping).

  1. Estimate scales: variogram \(\rightarrow\) \(\sigma_{\text{proc}}^2\), \(L_{50}\), \(L_{95}\).
  2. Couple scales: smooth predictors / choose grids according to \(R_{\text{micro}}\), \(R_{\text{local}}\).
  3. Tune \(R^*\): block‑CV, U‑curve \(\rightarrow\) stable drift radius.
  4. Benchmark methods: compare OK/KED/GAM/RF/Trend/IDW/Voronoi at \(R^*\) (RMSE/MAE/Bias, document block size).
  5. Products: write maps/grids at \(R^*\) (and optionally \(L_{95}\)); report the error budget.

Key takeaway: The “smartest” algorithm doesn’t win — the one whose scale matches the process does.

11.1.2 I.5 Reading the outputs (tables & plots)

This section explains how to interpret the key tables and figures produced by the pipeline and how to turn them into a model choice and a scale statement.

11.1.2.1 1) Variogram & scale table (chunk scale-Ls)

  • What you see: Empirical variogram points/line, horizontal dotted line at the (structural) sill, and vertical dashed lines at L50 and L95.

  • How to read it:

    • Nugget (near‑zero intercept) ≈ measurement/microscale noise. A large nugget means close points differ substantially; no method can beat this noise floor.
    • Sill (plateau) ≈ total variance once pairs are effectively uncorrelated.
    • L50 / L95 ≈ pragmatic correlation distances (half vs. ~all structure spent). They are your scale anchors for smoothing radii, neighborhood ranges, and CV block sizes.

  • Quality checks:

    • If no clear plateau: trend/non‑stationarity is likely → consider a drift (elev/sun terms) or a larger domain.
    • If L95 is near the domain size: scales are long; block sizes should be generous to avoid leakage.
    • If the variogram is noisy at large lags: rely more on L50 and the U‑curve outcome.

11.1.2.2 2) U‑curve for tuned drift (chunk scale-tune)

  • What you see: A line plot of RMSE vs. smoothing radius R for KED under blocked CV.

  • Decision rule: R* is the radius with the lowest CV‑RMSE.

  • What shapes mean:

    • Left side high (too small R): drift carries microscale noise → overfitting → higher CV error.
    • Right side high (too large R): drift is oversmoothed → loses meaningful gradient → bias ↑.
    • Flat bottom/plateau: a range of R values are equivalent → pick the smallest R on the plateau for parsimony.

  • Edge cases: If the minimum sits at the search boundary, widen the R grid and re‑run; if still at the boundary, the field may be trend‑dominated or the covariate is weak.

11.1.2.3 3) LBO‑CV metrics table (res$metrics)

For each model (Voronoi, IDW, OK, KED, GAM, RF) we report:

  • RMSE (primary): square‑error penalty; most sensitive to outliers. Use this to rank models.
  • MAE: median‑like robustness; a useful tie‑breaker alongside RMSE.
  • Bias (mean error): systematic over/under‑prediction; prefer |Bias| close to 0.
  • R²: variance explained in held‑out blocks; interpret cautiously under spatial CV.
  • n: number of held‑out predictions contributing.

Choosing a winner:

  1. Rank by lowest RMSE under the tuned configuration.
  2. If RMSEs are within ~5–10%: prefer the model with lower MAE, lower |Bias|, and more stable block‑wise errors (see next point).
  3. If KED (R*) ≈ OK: the drift adds little; the covariate is weak or the process is long‑range. If GAM/RF wins, the relationship is nonlinear or interaction‑rich.

11.1.2.4 4) Block‑wise diagnostics

  • Block error boxes/scatter: Look for narrow distributions (stable across space). Large spread or outliers indicate location‑dependent performance.
  • Stability index (optional): CV_rmse = sd(RMSE_block) / mean(RMSE_block). Values < 0.25 are typically stable; > 0.4 suggests uneven performance.
  • Obs vs Pred scatter: Slope ~1 and tight cloud = good calibration; bowed patterns imply bias or missing drift terms.

11.1.2.5 5) Error budget table (make_simple_error_budget)

Three rows show how error decreases as structure is added and matched:

  • Baseline (OK): no drift; sets a structure‑free reference.
  • Add drift (KED base): uses raw covariate; improvement here quantifies signal in the covariate.
  • Scale‑match drift (KED R*): covariate smoothed at R*; additional gain isolates scale alignment. The Gain_vs_prev column is the incremental improvement at each step.

If KED base ~ KED R*, scale matching adds little (either the raw drift is already at a compatible scale, or the field is insensitive to R). If OK > KED base, the covariate may inject noise or the drift term is mis‑specified.

11.1.3 I.6 Deciding on the best model (and documenting the scale)

Use this practical, auditable rule set:

  1. Primary criterion: Lowest CV‑RMSE under blocked CV.
  2. Tie‑breakers: Lower MAE, smaller |Bias|, and better block‑stability.
  3. Parsimony: If multiple models tie, choose the simplest (OK/KED < GAM < RF).
  4. Scale sanity check: Report L50/L95 and verify that R* lies roughly in [L50, 1.5·L95]. If not, discuss why (e.g., strong trend, weak covariate, anisotropy).
  5. Reproducibility: Record the block size, R grid, winning R*, and the full metrics table.

11.1.4 I.7 Typical patterns & what they imply

  • High nugget, short L50: Expect modest absolute accuracy; prefer coarser R and conservative models. IDW/OK with tight neighborhoods can perform on par with KED.
  • Long L95, clear sill: Favor larger neighborhoods and smoother drifts; KED (R*) often dominates.
  • GAM/RF > KED: Nonlinear covariate effects or interactions (e.g., slope×aspect). Still align covariates to R* to avoid noise chasing.
  • OK ~ KED: Elevation (or chosen drift) is weak for this synthetic setup; consider enriching covariates (slope/aspect/TRI) at matched scales.

11.1.5 I.8 Checklist before you trust the numbers

  • Block size reflects correlation scale (≈ L95).
  • U‑curve scanned a broad enough R range; minimum not at boundary.
  • R* reported along with L50/L95.
  • Winner chosen by blocked CV (not random folds).
  • Bias near zero; residuals pattern‑free in space.
  • Figures/tables archived for reproducibility.